Discussion

An interesting result emerging from this work is that the rate of change of the CCN concentration in the cloud layer, and its influence on the cloud, is tied to the change in inversion layer height z_i. As the inversion height rises, one would expect the concentration of CCN in the boundary layer to increase with time until it reaches values greater than 150 cm^-3 as the boundary layer is progressively contaminated by the dirtier air. In the current simulations we have shown that the maximum concentration of CCN in the cloud layer never exceeds 60 cm^-3. Simple calculations given below provide an explanation for this (B. Stevens 1999, personal communication). By mass continuity the only way for the inversion to rise is if fluid from above the inversion is mixed into the boundary layer. Quantitatively, we initially have N_ccn=30 cm^-3 in the cloud layer and N_ccn=250 cm^-3 above the inversion height (Figure 1), while z_i rises from 460 m to 530 m by the end of the simulation (Figure 3a). The maximum change in the CCN concentration in the cloud layer due to entrainment for the 5-h long simulation will be approximately

$$N_{f}=({N_0}\times z_{i0}+N_i\times(z_{if} - z_{i0}))/z_{if} = N_i - \frac{z_{i0}}{z_{if}} (N_i - N_0)$$ (5)

where z_i0 and z_if denote initial and final values of z_i respectively; N₀ and N_i represent the initial values of N_ccn in the cloud and above inversion, and N_f represents the predicted value of N_ccn. By substituting the values produced in this study, we have $N_f=250 - \frac{460}{530}(250-30) =$ 59 cm^-3, which is in agreement with the number we have simulated in the high CCN case.

The simple calculation given above suggests that the effect of polluted layers aloft on underlying clouds will depend, at least in part, on the rate of increase in z_i. It is likely that the impact of the entrained CCN on the simulated cloud would be greater if z_i rises faster than it does in this study, provided that there are no significant changes in LWP. This result suggests that we may be able to predict changes in cloud optical properties either due to the contamination of clouds, as in this study, or due to dilution in cases where cleaner air over-lies dirtier air as seen in the ASTEX Lagrangian II experiment [Bretherton et al., 1995]. By combining the expression relating N_d to boundary layer evolution (Eq. 5) with Twomey's [1974] $\tau$ $\approx N_d^{1/3}$ scaling law and Bohren's [1987] simple relationship between cloud albedo A and optical depth (see equation 4 in Section 3 for detail), we can predict cloud albedo changes in response to changes in N_ccnfor a given LWC, or LWP.

Using $\tau_0 \approx N_0^{1/3}$ to represent for the constant CCN case and $\tau_f \approx N_f^{1/3}$ for the high CCN at the inversion case, we have, at constant LWP

$$\frac{\tau_f}{\tau_0} = \left ({N_f \over N_0} \right )^{1/3} \; .$$ (6)

From the two simulations we have N₀=30 cm^-3, N_f=60 cm^-3, $\tau_0 = 7.5$ at 200 min (see Figure 6b), and $\tau_f = 7.5 \times (60/30)^{1/3}=9.4$, which is very close to the number 9.5 calculated directly from the model. Similarly, we can derive the A_f from equation (4) and (5) as

$$\frac{A_f}{A_0} \approx \frac{\tau_f}{\tau_0}~ \frac{2 + (1-g)\tau_0}{2 + (1-g)\tau_f}$$ (7)

by substituting $\tau_0 = 7.5$, $\tau_f=9.4$ (from eqn (6), A₀=0.51, we have A_f = 0.58, which agrees very well with the number directly from the model.

Combining (5) and (6) we have a general expression predicting the change in cloud optical depth as a function of the change in z_iand the N_ccn differential across the boundary layer:

$${\tau_f \over \tau_0} = \left [ {N_i \over N_0} \left ( 1 - ... ...z_{if} } \right ) + {z_{io} \over z_{if} } \right ]^{1/3} \; .$$ (8)

It should be stressed that the above derivation proceeds under the premise that LWC or LWP remains close to constant and that neither entrainment, nor drizzle change the LWP significantly. In general, the extent to which this holds true will depend on the thermodynamic properties and aerosol characteristics of the air overlying the boundary layer. This study indicates that the estimation of z_i is central to boundary layer studies with implications for cloud dynamics, microphysics and cloud optical properties:

1. A boundary layer that rises rapidly and entrains warm, dry air will tend to produce a thinner, higher cloud with lower LWP. Entrainment of warm, dry free tropospheric air will also result in poor vertical mixing if the boundary layer gets too deep. It has been shown by many others that the same entrained air, under certain condition, can descend unstably, enhance TKE at cloud top, and promote more entrainment. Eventually this feedback mechanism (commonly referred to as cloud-top entrainment instability) can cause the dissipation of the cloud [e.g. Lilly, 1968; Randall, 1980; Deardorff, 1980; MacVean and Mason, 1990; Duynkerke, 1993; De Roode and Duynkerke, 1997]. Shallow boundary layers that do not rise, may produce significant drizzle which also reduces LWP. In some instances, a boundary layer that deepens slowly may dry sufficiently such that drizzle is suppressed and LWP is maintained approximately constant [Stevens, 1996b].

2. the rate at which free tropospheric aerosol particles (some subset of which are CCN) are entrained into the cloud layer will directly affect cloud optical depth and albedo, and indirectly affect boundary layer dynamics by modifying cloud-top cooling rates [e.g., Olsson et al., 1998] and modifying drizzle. The effect of drizzle on boundary layer dynamics has been considered in depth by Stevens et al. [1998].

3. A small fraction of the aerosol population (concentrations on the order of a few per liter to a few 10s per liter) may act as Ice Forming Nuclei (IFN). At high latitudes, the entrainment of free tropospheric air into stratiform clouds experiencing sub-zero temperatures could be strongly impacted by these IFN [Harrington et al., 1999; Jiang et al., 1999].

Note that the observed IFN sounding for this case exhibited a sharp increase in IFN concentrations above the inversion (Figure 9, courtesy of D. Rogers). Thus, had this case been cold enough for vigorous participation of the ice phase, we speculate that perturbation of the thermodynamic, dynamic, and radiative properties of the Arctic boundary layer by entrainment would have been much greater.